Can supervised learning be used to solve the inverted pendulum problem?
Of course, you can, and yes this has been researched and done before. By using supervised learning, you give the machine some data, and it will try to figure the best way to analyze and predict the next movement in case of an inverted pendulum problem.
I found a complete paper for this problem: Neural network control of an inverted pendulum on a cart by Valeri Mladenov et al.